1 00:00:00,000 --> 00:00:07,050 2 00:00:07,050 --> 00:00:12,550 PROFESSOR: OK, today's about inverse functions, which is a 3 00:00:12,550 --> 00:00:16,020 new way to create one function from another one. 4 00:00:16,020 --> 00:00:23,980 And the reason it's so important is that we want to-- 5 00:00:23,980 --> 00:00:28,810 this, the logarithm, is going to be the inverse function for 6 00:00:28,810 --> 00:00:30,270 e to the x. 7 00:00:30,270 --> 00:00:34,090 We can't live without e to the x on one side and the 8 00:00:34,090 --> 00:00:36,380 logarithm on the other side. 9 00:00:36,380 --> 00:00:39,680 So here's the idea of inverse functions. 10 00:00:39,680 --> 00:00:42,730 Well, here are the letters we use. 11 00:00:42,730 --> 00:00:45,390 Usually y is f of x. 12 00:00:45,390 --> 00:00:48,340 That's the standard letters. 13 00:00:48,340 --> 00:00:54,390 Then, for the inverse, I'm going to use f with a minus 14 00:00:54,390 --> 00:00:56,940 one above the line. 15 00:00:56,940 --> 00:01:01,560 And notice though, it will be x is f inverse of y. 16 00:01:01,560 --> 00:01:03,480 And let me show you why that is. 17 00:01:03,480 --> 00:01:06,310 Let's just remember what a function is. 18 00:01:06,310 --> 00:01:11,180 So function like f is, I take an x-- 19 00:01:11,180 --> 00:01:12,430 that's my input-- 20 00:01:12,430 --> 00:01:17,220 21 00:01:17,220 --> 00:01:24,585 then the function acts on that input and produces an output. 22 00:01:24,585 --> 00:01:29,980 23 00:01:29,980 --> 00:01:37,400 At some level, this is what a function is, a bunch of inputs 24 00:01:37,400 --> 00:01:40,300 and the corresponding outputs. 25 00:01:40,300 --> 00:01:42,890 OK, what's the inverse function? 26 00:01:42,890 --> 00:01:44,970 You can guess what's coming. 27 00:01:44,970 --> 00:01:47,350 I'll reverse those. 28 00:01:47,350 --> 00:01:52,870 For the inverse function, y will be the input. 29 00:01:52,870 --> 00:01:56,500 So y is now the input. 30 00:01:56,500 --> 00:01:59,710 What used to be the output is now the input-- 31 00:01:59,710 --> 00:02:01,820 just turning them around. 32 00:02:01,820 --> 00:02:07,400 Then the question is, what x did it come from? 33 00:02:07,400 --> 00:02:13,980 That x that y came from up here is the x that it goes to 34 00:02:13,980 --> 00:02:17,020 with the inverse function. 35 00:02:17,020 --> 00:02:18,370 So you see the point? 36 00:02:18,370 --> 00:02:21,230 X and y are just getting reversed. 37 00:02:21,230 --> 00:02:30,070 Let me do an example, because that's letters, and we need a 38 00:02:30,070 --> 00:02:31,580 first example. 39 00:02:31,580 --> 00:02:35,010 y is x squared. 40 00:02:35,010 --> 00:02:37,510 So that's my function f of x. 41 00:02:37,510 --> 00:02:41,140 f is this squaring function. 42 00:02:41,140 --> 00:02:47,090 If you give me x equals three, the output is y equals nine. 43 00:02:47,090 --> 00:02:50,850 Now, what's the inverse function? 44 00:02:50,850 --> 00:02:54,770 The inverse function, I want to find x from y. 45 00:02:54,770 --> 00:02:55,650 How do I do that? 46 00:02:55,650 --> 00:02:57,270 I take the square root. 47 00:02:57,270 --> 00:03:00,590 So the inverse function will be x is-- 48 00:03:00,590 --> 00:03:03,540 do you like to write square root of y or do you like to 49 00:03:03,540 --> 00:03:05,910 write y to the one half power? 50 00:03:05,910 --> 00:03:07,300 Both good-- 51 00:03:07,300 --> 00:03:12,990 That's the inverse function of the other, and, of course, 52 00:03:12,990 --> 00:03:16,930 before we said, if x was three y was nine. 53 00:03:16,930 --> 00:03:22,680 And, now, if y is nine, then x will come out to be the square 54 00:03:22,680 --> 00:03:26,090 root of nine, three. 55 00:03:26,090 --> 00:03:28,395 Oh, one small point-- well, not so small. 56 00:03:28,395 --> 00:03:32,750 57 00:03:32,750 --> 00:03:39,340 I was really staying there in this example with x greater or 58 00:03:39,340 --> 00:03:41,410 equal zero. 59 00:03:41,410 --> 00:03:46,440 I don't know want to allow x equal minus three. 60 00:03:46,440 --> 00:03:48,170 Well, why not? 61 00:03:48,170 --> 00:03:51,130 Because if I allowed x equal minus three as one of the 62 00:03:51,130 --> 00:03:55,110 inputs, if I extended the function x squared 63 00:03:55,110 --> 00:03:57,840 to go for all x's. 64 00:03:57,840 --> 00:04:03,440 So if x equal minus three was allowed input, then y would be 65 00:04:03,440 --> 00:04:05,140 the same answer, nine. 66 00:04:05,140 --> 00:04:07,580 So I would be getting nine from both 67 00:04:07,580 --> 00:04:09,660 three and minus three. 68 00:04:09,660 --> 00:04:14,520 And then, in the inverse, I wouldn't know which 69 00:04:14,520 --> 00:04:16,279 one to go back to. 70 00:04:16,279 --> 00:04:18,930 In the inverse, the input would be nine, but should the 71 00:04:18,930 --> 00:04:20,959 output be three or minus three. 72 00:04:20,959 --> 00:04:26,900 So the point is, our functions have to be-- 73 00:04:26,900 --> 00:04:31,360 one-to-one is a kind of nice expression that gives you the 74 00:04:31,360 --> 00:04:35,710 idea, one x for one y, one y for one x. 75 00:04:35,710 --> 00:04:40,730 And that means that they're graphs have to go steadily 76 00:04:40,730 --> 00:04:48,680 upwards or steadily downwards, but not down and up the way y 77 00:04:48,680 --> 00:04:52,930 equal x squared would if I went over all x's. 78 00:04:52,930 --> 00:04:58,520 Let me do some more examples before I come to the reason 79 00:04:58,520 --> 00:05:04,750 for this lecture, which is the exponential and the logarithm. 80 00:05:04,750 --> 00:05:08,650 And let's just look ahead to what will come near the end of 81 00:05:08,650 --> 00:05:09,870 the lecture. 82 00:05:09,870 --> 00:05:16,060 We know facts about the exponential, e to the x, and 83 00:05:16,060 --> 00:05:20,150 those facts, when I look at the inverse function, give me 84 00:05:20,150 --> 00:05:24,290 some different facts, important facts still, about 85 00:05:24,290 --> 00:05:25,130 the logarithm. 86 00:05:25,130 --> 00:05:27,200 And here is the most important one. 87 00:05:27,200 --> 00:05:33,750 That the log of a product of two numbers is the sum of the 88 00:05:33,750 --> 00:05:35,880 two logarithms, very important fact. 89 00:05:35,880 --> 00:05:39,710 That simple but important fact is what 90 00:05:39,710 --> 00:05:41,970 made logarithms famous. 91 00:05:41,970 --> 00:05:46,930 And it was the whole basis for the slide rule. 92 00:05:46,930 --> 00:05:48,870 Well, do you know what a slide rule is? 93 00:05:48,870 --> 00:05:50,340 Maybe you haven't ever seen one? 94 00:05:50,340 --> 00:05:52,140 Probably not. 95 00:05:52,140 --> 00:05:56,600 Everybody had them, and then suddenly nobody has any. 96 00:05:56,600 --> 00:05:59,690 But the point was, on a slide rule you had a little stick, 97 00:05:59,690 --> 00:06:00,585 another little stick-- 98 00:06:00,585 --> 00:06:02,730 I used to drop the thing-- 99 00:06:02,730 --> 00:06:08,940 And you push out log y, and then the second stick measures 100 00:06:08,940 --> 00:06:14,310 out log capital Y, and then you read off the answer of a 101 00:06:14,310 --> 00:06:15,180 multiplication. 102 00:06:15,180 --> 00:06:20,136 So you were able to multiply, but kind of inaccurately. 103 00:06:20,136 --> 00:06:21,386 So-- 104 00:06:21,386 --> 00:06:23,970 105 00:06:23,970 --> 00:06:27,110 But that doesn't mean that this 106 00:06:27,110 --> 00:06:28,800 logarithm isn't still important. 107 00:06:28,800 --> 00:06:31,500 It is, just not for slide rules. 108 00:06:31,500 --> 00:06:33,685 I promised two more examples. 109 00:06:33,685 --> 00:06:36,430 110 00:06:36,430 --> 00:06:38,820 Also this is a chance to think about functions. 111 00:06:38,820 --> 00:06:47,130 So what about the radius of a circle r and the area a? 112 00:06:47,130 --> 00:06:49,090 There is a function there. 113 00:06:49,090 --> 00:06:56,260 The input is r, and the area is pi r squared. 114 00:06:56,260 --> 00:07:05,230 So that's some function of r, input r, output a. 115 00:07:05,230 --> 00:07:07,850 Now, tell me the inverse function. 116 00:07:07,850 --> 00:07:11,470 The inverse function, I'm going to input a, and 117 00:07:11,470 --> 00:07:13,120 I want to get r. 118 00:07:13,120 --> 00:07:16,310 So the input for the inverse function, it's 119 00:07:16,310 --> 00:07:17,610 going to be like this. 120 00:07:17,610 --> 00:07:20,430 I have to solve that equation for r. 121 00:07:20,430 --> 00:07:25,260 122 00:07:25,260 --> 00:07:26,140 How do I do that? 123 00:07:26,140 --> 00:07:27,390 I divide by pi. 124 00:07:27,390 --> 00:07:30,440 125 00:07:30,440 --> 00:07:33,820 That gives me r squared. 126 00:07:33,820 --> 00:07:35,970 And then, just as there, I take the square 127 00:07:35,970 --> 00:07:37,910 root, and I have r. 128 00:07:37,910 --> 00:07:41,790 So that's the inverse function of-- 129 00:07:41,790 --> 00:07:43,940 is that a function of r? 130 00:07:43,940 --> 00:07:45,200 No way. 131 00:07:45,200 --> 00:07:46,570 The input is now a. 132 00:07:46,570 --> 00:07:49,840 This is a function of a. 133 00:07:49,840 --> 00:07:53,720 Divide by pi, take the square root, and you're back to r. 134 00:07:53,720 --> 00:07:56,260 Let me draw the pictures that go with that. 135 00:07:56,260 --> 00:07:59,680 Because the graph of a function and its inverse 136 00:07:59,680 --> 00:08:02,430 function are really quite neat. 137 00:08:02,430 --> 00:08:05,260 So do you know what the graph of a equals pi r 138 00:08:05,260 --> 00:08:06,300 squared would look like? 139 00:08:06,300 --> 00:08:11,190 Again, r is only going to be positive, and, now, area's 140 00:08:11,190 --> 00:08:12,710 only going to be positive. 141 00:08:12,710 --> 00:08:17,470 And the graph of pi r squared is a parabola. 142 00:08:17,470 --> 00:08:23,660 Say at r equal one, I reach area equal-- 143 00:08:23,660 --> 00:08:27,480 What would be the area if the radius is one? 144 00:08:27,480 --> 00:08:32,799 Plug it in the formula, the area is pi. 145 00:08:32,799 --> 00:08:33,960 That's that point. 146 00:08:33,960 --> 00:08:35,445 And those are all the other points. 147 00:08:35,445 --> 00:08:36,650 OK. 148 00:08:36,650 --> 00:08:43,309 So that's the graph, which was nothing new. 149 00:08:43,309 --> 00:08:46,335 The new graph is the graph of the inverse function. 150 00:08:46,335 --> 00:08:46,910 OK. 151 00:08:46,910 --> 00:08:48,320 What's up? 152 00:08:48,320 --> 00:08:59,590 This time the input is now a, and the output is r. 153 00:08:59,590 --> 00:09:02,900 If the area is 0, the radius is 0. 154 00:09:02,900 --> 00:09:05,320 If the area is pi-- 155 00:09:05,320 --> 00:09:07,750 Oh, look, I'm just going to take this, and 156 00:09:07,750 --> 00:09:08,830 it's going to go here. 157 00:09:08,830 --> 00:09:15,300 If the area is pi, what's the radius? 158 00:09:15,300 --> 00:09:17,100 Well, put it in the formula. 159 00:09:17,100 --> 00:09:20,030 If the area is pi, I have pi over pi, one, 160 00:09:20,030 --> 00:09:21,090 square root's one. 161 00:09:21,090 --> 00:09:22,770 The radius is one, of course it's one. 162 00:09:22,770 --> 00:09:26,430 163 00:09:26,430 --> 00:09:34,520 One there, so that's a point on the graph of the inverse 164 00:09:34,520 --> 00:09:39,050 function, of this square root of a over pi. 165 00:09:39,050 --> 00:09:42,850 And what's the rest of the graph. 166 00:09:42,850 --> 00:09:43,860 Does it look like that? 167 00:09:43,860 --> 00:09:45,200 No way. 168 00:09:45,200 --> 00:09:50,640 Everything is being flipped, you could say, Or you could 169 00:09:50,640 --> 00:09:56,860 say, a mirror image just turned over here. 170 00:09:56,860 --> 00:10:01,120 This thing, which started out like this, is now a square 171 00:10:01,120 --> 00:10:02,220 root function. 172 00:10:02,220 --> 00:10:05,690 The square root function climbs and comes around like 173 00:10:05,690 --> 00:10:11,400 that like it's a parabola this way, because that one was a 174 00:10:11,400 --> 00:10:14,390 parabola that way. 175 00:10:14,390 --> 00:10:17,120 All right, let me go on to a second example. 176 00:10:17,120 --> 00:10:22,480 But that point that these graphs just flip over the 45 177 00:10:22,480 --> 00:10:26,940 degree line, it's because y and x are getting switched. 178 00:10:26,940 --> 00:10:28,346 Let me do the second example. 179 00:10:28,346 --> 00:10:31,710 180 00:10:31,710 --> 00:10:32,960 What about temperature? 181 00:10:32,960 --> 00:10:36,200 182 00:10:36,200 --> 00:10:41,170 We could measure temperature in Fahrenheit, say f. 183 00:10:41,170 --> 00:10:47,260 Or we can measure it in centigrade or Celsius, say c. 184 00:10:47,260 --> 00:10:49,850 And what's the function? 185 00:10:49,850 --> 00:10:53,950 If I take f, I want to know c. 186 00:10:53,950 --> 00:10:59,340 The centigrade temperature is some function of f. 187 00:10:59,340 --> 00:11:01,030 Let me at the same time draw the 188 00:11:01,030 --> 00:11:03,470 picture so we can remember. 189 00:11:03,470 --> 00:11:06,060 So this is now the forward function. 190 00:11:06,060 --> 00:11:09,570 I'm creating f first, and then I'm going to create f inverse. 191 00:11:09,570 --> 00:11:10,860 OK. 192 00:11:10,860 --> 00:11:15,610 Do you remember the point how they're connected? 193 00:11:15,610 --> 00:11:18,000 Here is f. 194 00:11:18,000 --> 00:11:21,050 And we'll start with the freezing point of water. 195 00:11:21,050 --> 00:11:26,010 The freezing point of water is 32 Fahrenheit but is zero 196 00:11:26,010 --> 00:11:26,620 centigrade. 197 00:11:26,620 --> 00:11:30,410 That's why that system got created. 198 00:11:30,410 --> 00:11:35,220 So f equals 32, c equals zero. 199 00:11:35,220 --> 00:11:36,520 That's on the graph. 200 00:11:36,520 --> 00:11:39,110 201 00:11:39,110 --> 00:11:41,190 And then what's the other key point? 202 00:11:41,190 --> 00:11:48,140 The boiling point of water, so say, that's one f is 212. 203 00:11:48,140 --> 00:11:51,750 212 is the boiling point of water in Fahrenheit. 204 00:11:51,750 --> 00:11:56,150 And what's the boiling point in Celsius, centigrade? 205 00:11:56,150 --> 00:11:57,050 100-- 206 00:11:57,050 --> 00:11:58,860 I mean that system was-- 207 00:11:58,860 --> 00:12:03,440 I don't know where 32 and 212 came from, but 0 to 100 is 208 00:12:03,440 --> 00:12:08,470 pretty sensible, 0 and then 100. 209 00:12:08,470 --> 00:12:15,880 So that's the other point and then, actually, the graph is 210 00:12:15,880 --> 00:12:17,990 just straight line. 211 00:12:17,990 --> 00:12:22,150 212 00:12:22,150 --> 00:12:26,010 In fact, let's find the formula. 213 00:12:26,010 --> 00:12:28,950 What's the equation for that line? 214 00:12:28,950 --> 00:12:36,060 So I take f and I subtract 32, so that gets me at the right 215 00:12:36,060 --> 00:12:38,330 start, the right freezing point. 216 00:12:38,330 --> 00:12:42,810 And now I want to multiply by the right slope to get up to 217 00:12:42,810 --> 00:12:44,850 the right boiling point. 218 00:12:44,850 --> 00:12:48,840 So when I go over 180, I want to go up 100. 219 00:12:48,840 --> 00:12:52,400 So it's 100 over 180. 220 00:12:52,400 --> 00:12:57,200 That 180 was the 32 to 212. 221 00:12:57,200 --> 00:13:02,040 So the ratio of 100 to 180 that's, well, five to nine 222 00:13:02,040 --> 00:13:08,090 would be easier to write, so let me write five to nine. 223 00:13:08,090 --> 00:13:12,680 Is that OK for the graph of the original function? 224 00:13:12,680 --> 00:13:16,960 This is my function of f giving me c. 225 00:13:16,960 --> 00:13:24,620 Ready for the inverse function? 226 00:13:24,620 --> 00:13:26,700 Can you do this with me? 227 00:13:26,700 --> 00:13:29,950 c is now the input. 228 00:13:29,950 --> 00:13:32,600 f is now the output. 229 00:13:32,600 --> 00:13:36,090 What was c equals 0 and a 100, those were the 230 00:13:36,090 --> 00:13:39,920 key points for c. 231 00:13:39,920 --> 00:13:48,310 f equals 32 and 212 were they key points for f. 232 00:13:48,310 --> 00:13:50,780 This was on the graph, right? 233 00:13:50,780 --> 00:13:54,500 0 centigrade gives 32 Fahrenheit. 234 00:13:54,500 --> 00:13:56,850 100 centigrade matches 212. 235 00:13:56,850 --> 00:13:59,630 That's on the graph. 236 00:13:59,630 --> 00:14:03,967 And again, it's a line in between. 237 00:14:03,967 --> 00:14:05,217 Hoo! 238 00:14:05,217 --> 00:14:06,930 239 00:14:06,930 --> 00:14:09,310 My picture isn't so fantastic. 240 00:14:09,310 --> 00:14:12,110 That 212 really should be higher, and that line should 241 00:14:12,110 --> 00:14:14,810 be steeper. 242 00:14:14,810 --> 00:14:19,500 Let's see that from the formula for f inverse. 243 00:14:19,500 --> 00:14:23,325 What is going to be the steepness of the second line? 244 00:14:23,325 --> 00:14:24,575 OK. 245 00:14:24,575 --> 00:14:29,300 246 00:14:29,300 --> 00:14:33,820 Here I've done graphs with some numbers. 247 00:14:33,820 --> 00:14:35,220 Here I'm going to do algebra. 248 00:14:35,220 --> 00:14:37,830 249 00:14:37,830 --> 00:14:41,620 I mean, the point of algebra is-- 250 00:14:41,620 --> 00:14:44,510 you may have wondered, what was the point of algebra-- 251 00:14:44,510 --> 00:14:48,285 the point is to deal with all numbers at once. 252 00:14:48,285 --> 00:14:50,860 253 00:14:50,860 --> 00:14:55,020 I could write down some other numbers like, some in between 254 00:14:55,020 --> 00:15:00,160 number like 122 or something, probably corresponds to a 255 00:15:00,160 --> 00:15:03,420 centigrade of 50. 256 00:15:03,420 --> 00:15:07,880 But I can't live forever with numbers. 257 00:15:07,880 --> 00:15:11,330 I need symbols. 258 00:15:11,330 --> 00:15:13,560 That's where letters, algebra, comes in. 259 00:15:13,560 --> 00:15:15,770 So now, I'm going to do algebra. 260 00:15:15,770 --> 00:15:20,960 I want to get Fahrenheit out of centigrade. 261 00:15:20,960 --> 00:15:23,940 I want to solve this equation for f. 262 00:15:23,940 --> 00:15:25,160 How do you solve for f? 263 00:15:25,160 --> 00:15:30,920 Well, first thing is, get rid of that 5/9, multiply by 9/5. 264 00:15:30,920 --> 00:15:35,280 So now I have 9/5 of c. 265 00:15:35,280 --> 00:15:38,480 So that 5/9 is now over here. 266 00:15:38,480 --> 00:15:40,330 Now, I have an f minus 32. 267 00:15:40,330 --> 00:15:43,500 I want to bring the 32 over on to the c side. 268 00:15:43,500 --> 00:15:44,885 It'll come over as a plus. 269 00:15:44,885 --> 00:15:48,630 270 00:15:48,630 --> 00:15:52,980 So I've solved this equation for f, and that's told me, 271 00:15:52,980 --> 00:15:54,340 what's the inverse function. 272 00:15:54,340 --> 00:15:56,620 And you notice, it is a straight line. 273 00:15:56,620 --> 00:16:00,010 And what's it's slope by the way? 274 00:16:00,010 --> 00:16:05,630 Its slope is 9/5, where this had a slope of 5/9. 275 00:16:05,630 --> 00:16:10,120 That's going to happen, if you multiply. 276 00:16:10,120 --> 00:16:14,570 And sooner or later, in the inverse, you have to divide. 277 00:16:14,570 --> 00:16:18,930 So one slope is the reciprocal of the other slope. 278 00:16:18,930 --> 00:16:21,670 Well, it's especially easy when we see straight lines. 279 00:16:21,670 --> 00:16:23,860 OK. 280 00:16:23,860 --> 00:16:29,670 Now, are we ready for the real thing, meaning exponentials? 281 00:16:29,670 --> 00:16:31,170 OK. 282 00:16:31,170 --> 00:16:35,830 So I come back to this board, which tells me what I'm after. 283 00:16:35,830 --> 00:16:40,280 And raise that a little and go for it. 284 00:16:40,280 --> 00:16:43,880 285 00:16:43,880 --> 00:16:46,360 So, what am I saying here? 286 00:16:46,360 --> 00:16:51,950 I'm saying that the logarithm is going to be the inverse 287 00:16:51,950 --> 00:16:56,310 function of e to the x. 288 00:16:56,310 --> 00:16:59,510 And it's called the natural logarithm, and we use this 289 00:16:59,510 --> 00:17:03,055 letter n for natural. 290 00:17:03,055 --> 00:17:06,170 291 00:17:06,170 --> 00:17:10,170 Although, the truth is, that it's the only logarithm 292 00:17:10,170 --> 00:17:11,900 I ever think of. 293 00:17:11,900 --> 00:17:18,349 I would freely write L-O-G, because I would always mean 294 00:17:18,349 --> 00:17:20,640 this natural logarithm. 295 00:17:20,640 --> 00:17:25,450 296 00:17:25,450 --> 00:17:29,600 So I'm defining it as the inverse and probably a graph 297 00:17:29,600 --> 00:17:35,560 is the way to see what it looks like. 298 00:17:35,560 --> 00:17:39,570 So I need to graph of e to the x, and then, a 299 00:17:39,570 --> 00:17:40,660 graph of its inverse. 300 00:17:40,660 --> 00:17:45,870 And then, by the way, since we're doing calculus, our next 301 00:17:45,870 --> 00:17:50,340 lecture is going to find derivatives. 302 00:17:50,340 --> 00:17:52,920 We know the derivative of e to the x. 303 00:17:52,920 --> 00:17:54,340 It's e to the x. 304 00:17:54,340 --> 00:18:03,440 That's the remarkable property that we started with. 305 00:18:03,440 --> 00:18:07,280 Then we'll find the derivative of the 306 00:18:07,280 --> 00:18:10,010 log, the inverse function. 307 00:18:10,010 --> 00:18:14,060 And it will come out to be remarkable too, amazing, 308 00:18:14,060 --> 00:18:17,680 amazing, just what we needed, in fact. 309 00:18:17,680 --> 00:18:20,640 All right, but let's get an idea what that log looks like. 310 00:18:20,640 --> 00:18:25,070 I know you've seen logs before, but now we have this 311 00:18:25,070 --> 00:18:30,890 base e, e to the x that only comes in calculus. 312 00:18:30,890 --> 00:18:35,980 And let's graph it. 313 00:18:35,980 --> 00:18:41,810 So now my function of x is e to the x, and I 314 00:18:41,810 --> 00:18:43,060 want to graph it. 315 00:18:43,060 --> 00:18:46,070 316 00:18:46,070 --> 00:18:49,380 This is, of course, y. 317 00:18:49,380 --> 00:18:57,220 Actually, I realize, x can be negative or 318 00:18:57,220 --> 00:19:00,760 positive, no problem. 319 00:19:00,760 --> 00:19:02,530 But y-- 320 00:19:02,530 --> 00:19:05,810 e to the x, always comes out positive. 321 00:19:05,810 --> 00:19:09,090 The graph is going to be above-- 322 00:19:09,090 --> 00:19:11,120 Here's x. 323 00:19:11,120 --> 00:19:14,560 Let me draw the graph from 0 to one and say, 324 00:19:14,560 --> 00:19:15,810 back to minus one. 325 00:19:15,810 --> 00:19:18,980 326 00:19:18,980 --> 00:19:25,040 Then the graph is going to be above the axis here. 327 00:19:25,040 --> 00:19:26,760 Let's see, where is it? 328 00:19:26,760 --> 00:19:30,140 329 00:19:30,140 --> 00:19:33,240 When x is 0, what's y? 330 00:19:33,240 --> 00:19:39,970 y is e to the 0th power, which is one. 331 00:19:39,970 --> 00:19:41,890 e to the 0 is one. 332 00:19:41,890 --> 00:19:45,880 The exponential function starts at one, right there. 333 00:19:45,880 --> 00:19:48,960 That height is one. 334 00:19:48,960 --> 00:19:57,210 Now when x is one, y is e to the first power, which is e, 335 00:19:57,210 --> 00:19:59,880 about 2.78. 336 00:19:59,880 --> 00:20:05,000 So maybe up there, somewhere about here. 337 00:20:05,000 --> 00:20:10,600 So that height is e, corresponding to one. 338 00:20:10,600 --> 00:20:14,520 And what about when x is minus one? 339 00:20:14,520 --> 00:20:19,160 Then y is e to the minus one. 340 00:20:19,160 --> 00:20:21,650 e to the minus one is-- 341 00:20:21,650 --> 00:20:24,340 that minus says, divide. 342 00:20:24,340 --> 00:20:29,370 It's one over e to the first power, one over 2.78, 343 00:20:29,370 --> 00:20:33,330 something like 1/3 or so, something about there. 344 00:20:33,330 --> 00:20:41,290 And now, if I put in the other points here, the 345 00:20:41,290 --> 00:20:44,225 graph looks like that. 346 00:20:44,225 --> 00:20:48,220 And, actually, the reason I didn't go beyond x equal one 347 00:20:48,220 --> 00:20:58,450 is that it climbs so fast. e to the x takes off. 348 00:20:58,450 --> 00:21:04,990 It grows exponentially, if you can allow me to say that. 349 00:21:04,990 --> 00:21:08,710 Which reminds me, we don't often say, grows 350 00:21:08,710 --> 00:21:11,340 logarithmically. 351 00:21:11,340 --> 00:21:14,320 Well, let's see what grows logarithmically means. 352 00:21:14,320 --> 00:21:16,770 It means creeping along. 353 00:21:16,770 --> 00:21:22,220 If e to the x is zipping up real fast, then the log is 354 00:21:22,220 --> 00:21:24,550 going to go up only slowly. 355 00:21:24,550 --> 00:21:28,230 356 00:21:28,230 --> 00:21:29,720 So I want the inverse function. 357 00:21:29,720 --> 00:21:33,060 Of course, this graph continues, 358 00:21:33,060 --> 00:21:35,250 gets very, very small. 359 00:21:35,250 --> 00:21:36,830 It continues up here. 360 00:21:36,830 --> 00:21:42,380 It gets very, very big but keeps going. 361 00:21:42,380 --> 00:21:48,700 Now, ready for x equals log y. 362 00:21:48,700 --> 00:21:55,130 And remember, I'm going to draw its picture, and I'm 363 00:21:55,130 --> 00:22:01,430 defining that function as the inverse function of the one we 364 00:22:01,430 --> 00:22:04,940 have. So I'm not going to give a new 365 00:22:04,940 --> 00:22:07,750 definition, a new function. 366 00:22:07,750 --> 00:22:17,470 It's defined by being the inverse function. 367 00:22:17,470 --> 00:22:19,460 That's what it is. 368 00:22:19,460 --> 00:22:23,370 But now we know, from experience with two graphs, we 369 00:22:23,370 --> 00:22:26,170 know what its graph is going to look like. 370 00:22:26,170 --> 00:22:29,130 So x is now going to be this graph, and y is 371 00:22:29,130 --> 00:22:29,730 going to be that one. 372 00:22:29,730 --> 00:22:32,150 So y only is positive. 373 00:22:32,150 --> 00:22:35,910 We can only take the log of positive numbers. 374 00:22:35,910 --> 00:22:38,680 375 00:22:38,680 --> 00:22:41,270 The log of a negative number, that's something imaginary, 376 00:22:41,270 --> 00:22:42,520 we're not touching that. 377 00:22:42,520 --> 00:22:45,400 378 00:22:45,400 --> 00:22:55,990 The log can come out positive, zero, negative. 379 00:22:55,990 --> 00:22:57,150 x could be anything. 380 00:22:57,150 --> 00:23:00,290 Here is x. 381 00:23:00,290 --> 00:23:02,630 Let's put in the points we know. 382 00:23:02,630 --> 00:23:08,020 They'll be the same three points as here, but you see 383 00:23:08,020 --> 00:23:15,210 that x axis is now vertical, the y axis is now horizontal. 384 00:23:15,210 --> 00:23:18,330 I put in these points, now let me put in-- 385 00:23:18,330 --> 00:23:21,050 So what's the thing? 386 00:23:21,050 --> 00:23:26,370 When y is 0, what's x? 387 00:23:26,370 --> 00:23:28,490 Yeah, can you get that one? 388 00:23:28,490 --> 00:23:29,740 What's the log--oh, no. 389 00:23:29,740 --> 00:23:32,620 390 00:23:32,620 --> 00:23:34,690 y doesn't make it to 0. 391 00:23:34,690 --> 00:23:38,020 When y is one, that's what I meant to say. 392 00:23:38,020 --> 00:23:43,250 When y is one, what's the log of one? 393 00:23:43,250 --> 00:23:44,810 What is the log of one? 394 00:23:44,810 --> 00:23:47,730 That's a key point here, and we see it. 395 00:23:47,730 --> 00:23:50,580 We got y equal one when x is 0. 396 00:23:50,580 --> 00:23:56,930 The logarithm of one is-- 397 00:23:56,930 --> 00:23:59,980 so when y is one-- 398 00:23:59,980 --> 00:24:01,540 is that right? 399 00:24:01,540 --> 00:24:05,170 Logarithm of one, let's put in that one. 400 00:24:05,170 --> 00:24:07,410 The logarithm of one is 0. 401 00:24:07,410 --> 00:24:10,150 That's a point on our curve. 402 00:24:10,150 --> 00:24:11,770 That's a point on our curve. 403 00:24:11,770 --> 00:24:16,690 This point flips down to this point. 404 00:24:16,690 --> 00:24:20,620 Can I just remind myself, because you saw me hesitating, 405 00:24:20,620 --> 00:24:23,590 that the log of one is 0. 406 00:24:23,590 --> 00:24:26,910 It's nice to have a couple of numbers. 407 00:24:26,910 --> 00:24:28,460 Then what are the other ones I want? 408 00:24:28,460 --> 00:24:31,260 I want to know the log of e, and I want to 409 00:24:31,260 --> 00:24:34,980 know the log of 1/e. 410 00:24:34,980 --> 00:24:38,570 And what are those logarithms? 411 00:24:38,570 --> 00:24:41,160 I could look over here. 412 00:24:41,160 --> 00:24:44,860 They're going to be one and minus one. 413 00:24:44,860 --> 00:24:49,000 But let's just begin to get the idea of the log. 414 00:24:49,000 --> 00:24:54,390 The log is the exponent. 415 00:24:54,390 --> 00:24:56,860 That's what you should say to yourself all the time. 416 00:24:56,860 --> 00:24:58,290 What is the logarithm? 417 00:24:58,290 --> 00:25:03,360 The logarithm is the exponent in the original. 418 00:25:03,360 --> 00:25:08,510 So here the exponent is one, so the log is one. 419 00:25:08,510 --> 00:25:10,930 What's the exponent there? 420 00:25:10,930 --> 00:25:14,480 One over e, that's e to the minus one power. 421 00:25:14,480 --> 00:25:18,260 That's log of e to the minus one power. 422 00:25:18,260 --> 00:25:19,890 And what is that logarithm? 423 00:25:19,890 --> 00:25:23,260 It's the exponent minus one. 424 00:25:23,260 --> 00:25:25,030 Let me plot those points. 425 00:25:25,030 --> 00:25:27,030 Here is e. 426 00:25:27,030 --> 00:25:30,350 So there is one, here is e, here is 1/e. 427 00:25:30,350 --> 00:25:34,760 428 00:25:34,760 --> 00:25:36,750 The logarithm of one was 0. 429 00:25:36,750 --> 00:25:38,430 That point's on my graph. 430 00:25:38,430 --> 00:25:40,800 The logarithm of e is one. 431 00:25:40,800 --> 00:25:42,750 This point's on my graph. 432 00:25:42,750 --> 00:25:45,760 The logarithm of 1/e is minus one. 433 00:25:45,760 --> 00:25:55,040 The curve is coming up like that but bending down just the 434 00:25:55,040 --> 00:25:59,420 way this curve was bending up. 435 00:25:59,420 --> 00:26:03,970 And if I continue the curve, the logarithm would get more 436 00:26:03,970 --> 00:26:04,720 and more negative. 437 00:26:04,720 --> 00:26:11,540 It's headed down there, but y is never allowed to be 0. 438 00:26:11,540 --> 00:26:14,320 Headed up here, what happens? 439 00:26:14,320 --> 00:26:18,450 The log of a million, the log of a trillion-- 440 00:26:18,450 --> 00:26:22,960 I mean, we can deal with the national debt, just take its 441 00:26:22,960 --> 00:26:31,260 log, because it climbs so slowly. 442 00:26:31,260 --> 00:26:32,770 Notice it kept climbing. 443 00:26:32,770 --> 00:26:37,700 It doesn't peak off here. 444 00:26:37,700 --> 00:26:40,630 That's a little farther than I intended to draw it. 445 00:26:40,630 --> 00:26:45,090 That's pretty far out on the y axis, but not very 446 00:26:45,090 --> 00:26:46,420 high on the x axis. 447 00:26:46,420 --> 00:26:50,810 Logarithms of big numbers are quite small numbers. 448 00:26:50,810 --> 00:26:56,680 And that's actually why, as we'll see, 449 00:26:56,680 --> 00:26:58,490 people use log paper. 450 00:26:58,490 --> 00:27:00,640 They draw a log-log graphs. 451 00:27:00,640 --> 00:27:04,570 That's to get big numbers on to the graph by 452 00:27:04,570 --> 00:27:06,480 dealing with logs. 453 00:27:06,480 --> 00:27:12,210 So that's what I want to say about the logarithm. 454 00:27:12,210 --> 00:27:20,070 Except, to come back to these two key facts, 455 00:27:20,070 --> 00:27:23,010 especially the first one. 456 00:27:23,010 --> 00:27:26,780 Can I find space for that first one? 457 00:27:26,780 --> 00:27:32,990 458 00:27:32,990 --> 00:27:36,130 So y is e to the x, as always. 459 00:27:36,130 --> 00:27:46,770 Capital Y would be e to the capital X. And now, the 460 00:27:46,770 --> 00:27:50,450 interesting property is what happens if I multiply. 461 00:27:50,450 --> 00:27:53,010 What happens if I multiply y times Y? 462 00:27:53,010 --> 00:27:55,870 463 00:27:55,870 --> 00:28:01,780 I have, that's e to the x times e to the X. That's what 464 00:28:01,780 --> 00:28:04,210 the little y and big y were. 465 00:28:04,210 --> 00:28:09,930 Now we're ready to use the crucial property of the 466 00:28:09,930 --> 00:28:11,180 exponential curve. 467 00:28:11,180 --> 00:28:13,660 468 00:28:13,660 --> 00:28:16,670 I'm asking you, because you have to know this. 469 00:28:16,670 --> 00:28:19,905 What is e to the x times e to the capital X? 470 00:28:19,905 --> 00:28:25,520 471 00:28:25,520 --> 00:28:29,440 Suppose x was two and capital X was three? 472 00:28:29,440 --> 00:28:35,590 Then I have e times e, e squared, multiplying e times e 473 00:28:35,590 --> 00:28:39,570 times e, three e's. 474 00:28:39,570 --> 00:28:40,370 What do I have? 475 00:28:40,370 --> 00:28:45,520 I've got e times e, time e times e time e. 476 00:28:45,520 --> 00:28:47,715 All together five e's are getting multiplied. 477 00:28:47,715 --> 00:28:50,420 478 00:28:50,420 --> 00:28:55,440 I just add the exponents. 479 00:28:55,440 --> 00:28:58,870 That's the big rule for the exponential. 480 00:28:58,870 --> 00:29:04,360 If I multiply exponentials, I add the exponents. 481 00:29:04,360 --> 00:29:09,040 Now, I just want to convert that to a rule for logarithms. 482 00:29:09,040 --> 00:29:10,950 I'm going to do the inverse function. 483 00:29:10,950 --> 00:29:14,990 I'm going to take the log of both sides, and, I hope, we're 484 00:29:14,990 --> 00:29:16,620 going to get the right thing. 485 00:29:16,620 --> 00:29:22,790 The logarithm of this is-- 486 00:29:22,790 --> 00:29:27,910 Well, what's the logarithm of this result? 487 00:29:27,910 --> 00:29:30,610 It's the exponent. 488 00:29:30,610 --> 00:29:34,820 The logarithm of this number is that. 489 00:29:34,820 --> 00:29:39,120 Just the way the logarithm of e to that number was the one. 490 00:29:39,120 --> 00:29:42,260 The logarithm of e to that number was the minus one. 491 00:29:42,260 --> 00:29:47,470 The logarithm of this number is the exponent x plus capital 492 00:29:47,470 --> 00:29:53,380 X. And finally what is little x? 493 00:29:53,380 --> 00:29:55,740 Well, don't forget where it came from. 494 00:29:55,740 --> 00:30:02,480 Little x is the exponent for y, so little x is log y. 495 00:30:02,480 --> 00:30:09,810 And capital X is log capital Y. 496 00:30:09,810 --> 00:30:12,900 Bunch of symbols on that board. 497 00:30:12,900 --> 00:30:16,670 And the last line is the one that we were shooting for. 498 00:30:16,670 --> 00:30:21,730 The logarithm of y times Y is the sum of the logs. 499 00:30:21,730 --> 00:30:25,360 500 00:30:25,360 --> 00:30:27,585 Because this guy is also important-- 501 00:30:27,585 --> 00:30:31,940 502 00:30:31,940 --> 00:30:35,570 Maybe I don't even give a proof. 503 00:30:35,570 --> 00:30:39,150 Because it's intimately related to this one, why don't 504 00:30:39,150 --> 00:30:40,720 I just see it. 505 00:30:40,720 --> 00:30:42,545 What would be the log of y squared? 506 00:30:42,545 --> 00:30:51,370 507 00:30:51,370 --> 00:30:53,650 Actually, we already-- here. 508 00:30:53,650 --> 00:30:57,750 If I wanted little y squared, what should I do? 509 00:30:57,750 --> 00:31:00,110 I can get that answer from what I've done. 510 00:31:00,110 --> 00:31:04,660 The log of little y squared, I just take big Y to be the same 511 00:31:04,660 --> 00:31:08,180 as little y, I take big X to be the same as little x, and 512 00:31:08,180 --> 00:31:10,190 I've got the log of y squared. 513 00:31:10,190 --> 00:31:14,030 Then is x plus x, two x's-- 514 00:31:14,030 --> 00:31:16,910 515 00:31:16,910 --> 00:31:18,810 but x is the log of y. 516 00:31:18,810 --> 00:31:23,660 517 00:31:23,660 --> 00:31:29,530 If you square a number, you only double its log. 518 00:31:29,530 --> 00:31:37,860 You're again seeing why these numbers can grow very quickly 519 00:31:37,860 --> 00:31:41,710 by squaring and squaring and squaring, but the logarithms 520 00:31:41,710 --> 00:31:46,940 only grow by multiplying by two, only going up slowly. 521 00:31:46,940 --> 00:31:54,150 And then the general result would be for any power, not 522 00:31:54,150 --> 00:31:57,840 just n equals two, not just n equals a whole number, not 523 00:31:57,840 --> 00:32:05,450 just n equals positive numbers, but all n, will be-- 524 00:32:05,450 --> 00:32:07,670 I'll have n of these-- 525 00:32:07,670 --> 00:32:11,470 so I'll have n logarithm of y. 526 00:32:11,470 --> 00:32:19,060 So that's a closely related property that takes the same y 527 00:32:19,060 --> 00:32:19,740 to different powers. 528 00:32:19,740 --> 00:32:20,990 OK. 529 00:32:20,990 --> 00:32:23,720 530 00:32:23,720 --> 00:32:29,170 Lots of symbols today, but you had to get that logarithm 531 00:32:29,170 --> 00:32:32,420 function straight before we can take its derivative. 532 00:32:32,420 --> 00:32:34,800 Can I tell you what its derivative is? 533 00:32:34,800 --> 00:32:37,410 Would you like to know in advance? 534 00:32:37,410 --> 00:32:38,660 The derivative-- 535 00:32:38,660 --> 00:32:41,460 536 00:32:41,460 --> 00:32:43,080 I don't know if I should tell you. 537 00:32:43,080 --> 00:32:49,160 The derivative of log y, the derivative of this log 538 00:32:49,160 --> 00:32:52,600 function, turns out to be 1/y. 539 00:32:52,600 --> 00:32:56,730 540 00:32:56,730 --> 00:32:58,360 Isn't that nice. 541 00:32:58,360 --> 00:33:01,950 A really good answer coming from this function that we 542 00:33:01,950 --> 00:33:04,040 created as an inverse function. 543 00:33:04,040 --> 00:33:06,750 And I'll just say here that now 544 00:33:06,750 --> 00:33:09,140 we've created the function. 545 00:33:09,140 --> 00:33:11,220 We've got it. 546 00:33:11,220 --> 00:33:15,680 Then I don't mind if you give it a different letter, give it 547 00:33:15,680 --> 00:33:16,850 another name. 548 00:33:16,850 --> 00:33:19,590 Well, I hope you keep its name log. 549 00:33:19,590 --> 00:33:21,190 Most people use that name. 550 00:33:21,190 --> 00:33:22,810 But you could use a different letter. 551 00:33:22,810 --> 00:33:26,910 I'm perfectly happy for you to write this as the derivative 552 00:33:26,910 --> 00:33:31,510 of log x is 1/x. 553 00:33:31,510 --> 00:33:35,340 Between that and that, I've just changed letters. 554 00:33:35,340 --> 00:33:40,820 That was like after the real thinking of this lecture, 555 00:33:40,820 --> 00:33:45,530 which was the when x was an input and y was an output, and 556 00:33:45,530 --> 00:33:48,610 I really needed two different letters. 557 00:33:48,610 --> 00:33:50,120 OK, good, that's inverse functions. 558 00:33:50,120 --> 00:33:52,226 Thank you. 559 00:33:52,226 --> 00:33:54,480 ANNOUNCER: This has been a production of MIT 560 00:33:54,480 --> 00:33:56,870 OpenCourseWare and Gilbert Strang. 561 00:33:56,870 --> 00:33:59,150 Funding for this video was provided by the Lord 562 00:33:59,150 --> 00:34:00,360 Foundation. 563 00:34:00,360 --> 00:34:03,490 To help OCW continue to provide free and open access 564 00:34:03,490 --> 00:34:06,570 to MIT courses please make a donation at 565 00:34:06,570 --> 00:34:08,130 ocw.mit.edu/donate. 566 00:34:08,130 --> 00:34:10,274